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Theorem clmnegsubdi2 24550

Description: Distribution of negative over vector subtraction. (Contributed by NM, 6-Aug-2007.) (Revised by AV, 29-Sep-2021.)

**Hypotheses**
Ref	Expression
clmpm1dir.v	⊢ 𝑉 = (Base‘𝑊)
clmpm1dir.s	⊢ · = ( ·_𝑠 ‘𝑊)
clmpm1dir.a	⊢ + = (+_g‘𝑊)

**Assertion**
Ref	Expression
clmnegsubdi2	⊢ ((𝑊 ∈ ℂMod ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (-1 · (𝐴 + (-1 · 𝐵))) = (𝐵 + (-1 · 𝐴)))

**Proof of Theorem clmnegsubdi2**
Step	Hyp	Ref	Expression
1		simp1 1136	. . 3 ⊢ ((𝑊 ∈ ℂMod ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → 𝑊 ∈ ℂMod)
2		eqid 2731	. . . . 5 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊)
3		eqid 2731	. . . . 5 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
4	2, 3	clmneg1 24527	. . . 4 ⊢ (𝑊 ∈ ℂMod → -1 ∈ (Base‘(Scalar‘𝑊)))
5	4	3ad2ant1 1133	. . 3 ⊢ ((𝑊 ∈ ℂMod ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → -1 ∈ (Base‘(Scalar‘𝑊)))
6		simp2 1137	. . 3 ⊢ ((𝑊 ∈ ℂMod ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → 𝐴 ∈ 𝑉)
7		simpl 483	. . . . 5 ⊢ ((𝑊 ∈ ℂMod ∧ 𝐵 ∈ 𝑉) → 𝑊 ∈ ℂMod)
8	4	adantr 481	. . . . 5 ⊢ ((𝑊 ∈ ℂMod ∧ 𝐵 ∈ 𝑉) → -1 ∈ (Base‘(Scalar‘𝑊)))
9		simpr 485	. . . . 5 ⊢ ((𝑊 ∈ ℂMod ∧ 𝐵 ∈ 𝑉) → 𝐵 ∈ 𝑉)
10		clmpm1dir.v	. . . . . 6 ⊢ 𝑉 = (Base‘𝑊)
11		clmpm1dir.s	. . . . . 6 ⊢ · = ( ·_𝑠 ‘𝑊)
12	10, 2, 11, 3	clmvscl 24533	. . . . 5 ⊢ ((𝑊 ∈ ℂMod ∧ -1 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝐵 ∈ 𝑉) → (-1 · 𝐵) ∈ 𝑉)
13	7, 8, 9, 12	syl3anc 1371	. . . 4 ⊢ ((𝑊 ∈ ℂMod ∧ 𝐵 ∈ 𝑉) → (-1 · 𝐵) ∈ 𝑉)
14	13	3adant2 1131	. . 3 ⊢ ((𝑊 ∈ ℂMod ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (-1 · 𝐵) ∈ 𝑉)
15		clmpm1dir.a	. . . 4 ⊢ + = (+_g‘𝑊)
16	10, 2, 11, 3, 15	clmvsdi 24537	. . 3 ⊢ ((𝑊 ∈ ℂMod ∧ (-1 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝐴 ∈ 𝑉 ∧ (-1 · 𝐵) ∈ 𝑉)) → (-1 · (𝐴 + (-1 · 𝐵))) = ((-1 · 𝐴) + (-1 · (-1 · 𝐵))))
17	1, 5, 6, 14, 16	syl13anc 1372	. 2 ⊢ ((𝑊 ∈ ℂMod ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (-1 · (𝐴 + (-1 · 𝐵))) = ((-1 · 𝐴) + (-1 · (-1 · 𝐵))))
18	10, 11, 15	clmnegneg 24549	. . . 4 ⊢ ((𝑊 ∈ ℂMod ∧ 𝐵 ∈ 𝑉) → (-1 · (-1 · 𝐵)) = 𝐵)
19	18	3adant2 1131	. . 3 ⊢ ((𝑊 ∈ ℂMod ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (-1 · (-1 · 𝐵)) = 𝐵)
20	19	oveq2d 7409	. 2 ⊢ ((𝑊 ∈ ℂMod ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → ((-1 · 𝐴) + (-1 · (-1 · 𝐵))) = ((-1 · 𝐴) + 𝐵))
21		clmabl 24514	. . . 4 ⊢ (𝑊 ∈ ℂMod → 𝑊 ∈ Abel)
22	21	3ad2ant1 1133	. . 3 ⊢ ((𝑊 ∈ ℂMod ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → 𝑊 ∈ Abel)
23		simpl 483	. . . . 5 ⊢ ((𝑊 ∈ ℂMod ∧ 𝐴 ∈ 𝑉) → 𝑊 ∈ ℂMod)
24	4	adantr 481	. . . . 5 ⊢ ((𝑊 ∈ ℂMod ∧ 𝐴 ∈ 𝑉) → -1 ∈ (Base‘(Scalar‘𝑊)))
25		simpr 485	. . . . 5 ⊢ ((𝑊 ∈ ℂMod ∧ 𝐴 ∈ 𝑉) → 𝐴 ∈ 𝑉)
26	10, 2, 11, 3	clmvscl 24533	. . . . 5 ⊢ ((𝑊 ∈ ℂMod ∧ -1 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝐴 ∈ 𝑉) → (-1 · 𝐴) ∈ 𝑉)
27	23, 24, 25, 26	syl3anc 1371	. . . 4 ⊢ ((𝑊 ∈ ℂMod ∧ 𝐴 ∈ 𝑉) → (-1 · 𝐴) ∈ 𝑉)
28	27	3adant3 1132	. . 3 ⊢ ((𝑊 ∈ ℂMod ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (-1 · 𝐴) ∈ 𝑉)
29		simp3 1138	. . 3 ⊢ ((𝑊 ∈ ℂMod ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → 𝐵 ∈ 𝑉)
30	10, 15	ablcom 19631	. . 3 ⊢ ((𝑊 ∈ Abel ∧ (-1 · 𝐴) ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → ((-1 · 𝐴) + 𝐵) = (𝐵 + (-1 · 𝐴)))
31	22, 28, 29, 30	syl3anc 1371	. 2 ⊢ ((𝑊 ∈ ℂMod ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → ((-1 · 𝐴) + 𝐵) = (𝐵 + (-1 · 𝐴)))
32	17, 20, 31	3eqtrd 2775	1 ⊢ ((𝑊 ∈ ℂMod ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (-1 · (𝐴 + (-1 · 𝐵))) = (𝐵 + (-1 · 𝐴)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ‘cfv 6532 (class class class)co 7393 1c1 11093 -cneg 11427 Basecbs 17126 +_gcplusg 17179 Scalarcsca 17182 ·_𝑠 cvsca 17183 Abelcabl 19613 ℂModcclm 24507

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2702 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7708 ax-cnex 11148 ax-resscn 11149 ax-1cn 11150 ax-icn 11151 ax-addcl 11152 ax-addrcl 11153 ax-mulcl 11154 ax-mulrcl 11155 ax-mulcom 11156 ax-addass 11157 ax-mulass 11158 ax-distr 11159 ax-i2m1 11160 ax-1ne0 11161 ax-1rid 11162 ax-rnegex 11163 ax-rrecex 11164 ax-cnre 11165 ax-pre-lttri 11166 ax-pre-lttrn 11167 ax-pre-ltadd 11168 ax-pre-mulgt0 11169 ax-addf 11171 ax-mulf 11172

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4523 df-pw 4598 df-sn 4623 df-pr 4625 df-tp 4627 df-op 4629 df-uni 4902 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6289 df-ord 6356 df-on 6357 df-lim 6358 df-suc 6359 df-iota 6484 df-fun 6534 df-fn 6535 df-f 6536 df-f1 6537 df-fo 6538 df-f1o 6539 df-fv 6540 df-riota 7349 df-ov 7396 df-oprab 7397 df-mpo 7398 df-om 7839 df-1st 7957 df-2nd 7958 df-frecs 8248 df-wrecs 8279 df-recs 8353 df-rdg 8392 df-1o 8448 df-er 8686 df-en 8923 df-dom 8924 df-sdom 8925 df-fin 8926 df-pnf 11232 df-mnf 11233 df-xr 11234 df-ltxr 11235 df-le 11236 df-sub 11428 df-neg 11429 df-nn 12195 df-2 12257 df-3 12258 df-4 12259 df-5 12260 df-6 12261 df-7 12262 df-8 12263 df-9 12264 df-n0 12455 df-z 12541 df-dec 12660 df-uz 12805 df-fz 13467 df-seq 13949 df-struct 17062 df-sets 17079 df-slot 17097 df-ndx 17109 df-base 17127 df-ress 17156 df-plusg 17192 df-mulr 17193 df-starv 17194 df-tset 17198 df-ple 17199 df-ds 17201 df-unif 17202 df-0g 17369 df-mgm 18543 df-sgrp 18592 df-mnd 18603 df-grp 18797 df-minusg 18798 df-mulg 18923 df-subg 18975 df-cmn 19614 df-abl 19615 df-mgp 19947 df-ur 19964 df-ring 20016 df-cring 20017 df-subrg 20310 df-lmod 20422 df-cnfld 20879 df-clm 24508

This theorem is referenced by: (None)

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